Diaschismic family - Revision history

FloraC: Recategorize

2026-08-14T11:54:02Z

Recategorize

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			[[Category:Diaschismic family\| ]] <!-- main article -->
	[[Category:Temperament families]]		[[Category:Temperament families]]
	[[Category:~~Diaschismic family\| ]] <!~~-~~- main article -->~~		[[Category:Catalogs of rank-2 temperaments]]
	~~[[Category:Rank~~ 2]]

FloraC: Fix a link

2026-07-22T15:47:45Z

Fix a link

← Older revision		Revision as of 15:47, 22 July 2026
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	==== Subgroup extensions ====		==== Subgroup extensions ====
	Since the diaschisma factors into ([[256/255]])<sup>2</sup>([[289/288]]) in the 17-limit, it extends naturally to the 2.3.5.17 subgroup as ''srutal archagall'', considered in [[#Subgroup extensions]]. The [[S-expression]]-based comma list of this temperament is {[[256/255\|S16]], [[289/288\|S17]]}.		Since the diaschisma factors into ([[256/255]])<sup>2</sup>([[289/288]]) in the 17-limit, it extends naturally to the 2.3.5.17 subgroup as ''srutal archagall'', considered in [[#Subgroup extensions_2\|#Subgroup extensions]]. The [[S-expression]]-based comma list of this temperament is {[[256/255\|S16]], [[289/288\|S17]]}.

	== Septimal diaschismic ==		== Septimal diaschismic ==

FloraC: /* Subgroup extensions */ move description here, adapted for this family

2026-07-22T15:46:29Z

Subgroup extensions: move description here, adapted for this family

← Older revision		Revision as of 15:46, 22 July 2026
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	=== Srutal archagall (2.3.5.17) ===		=== Srutal archagall (2.3.5.17) ===
	{{See also \| Fiventeen }}		{{See also \| Fiventeen }}

			This extension of 5-limit diaschismic adds prime 17 and which with respect to [[MVP archagall]] is able to express the harmonics [[75/1\|75]] and [[85/1\|85]] in their appropriate prime subgroup. It achieves this by equating [[85/64]] with [[4/3]] by tempering out their difference of [[256/255]] (S16). Therefore it also tempers out [[289/288]] (S17) and thus equates [[17/15]] with [[9/8]] due to tempering out [[136/135]] (S16⋅S17). It could be described as the 10 & 12 temperament with strong emphasis on 12edo being the better tuning on the 2.3.5.17 subgroup, implying ideal tunings of 34edo, 46edo or 80edo.

	Subgroup: 2.3.5.17		Subgroup: 2.3.5.17

FloraC: /* Sruti */ + redirect notice

2026-07-20T09:48:35Z

Sruti: + redirect notice

← Older revision		Revision as of 09:48, 20 July 2026
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	== Sruti ==		== Sruti ==
			{{Redirect\|Sruti\|the tuning concept in Indian music\|Shruti}}

	Sruti tempers out 19683/19600, setting itself up as a [[hemipyth]] temperament. It has the same semi-octave period as diaschismic, but the generator can be taken as a neutral third or a hemitwelfth. The temperament can be described as {{nowrap\| 24 & 34d }}; its ploidacot is diploid dicot. [[58edo]] may be recommended as a tuning.		Sruti tempers out 19683/19600, setting itself up as a [[hemipyth]] temperament. It has the same semi-octave period as diaschismic, but the generator can be taken as a neutral third or a hemitwelfth. The temperament can be described as {{nowrap\| 24 & 34d }}; its ploidacot is diploid dicot. [[58edo]] may be recommended as a tuning.

FloraC: Text replacement - "octave" to "octave"

2026-07-14T14:15:53Z

Text replacement - "octave" to "octave"

← Older revision		Revision as of 14:15, 14 July 2026
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	{{Main\| Diaschismic }}		{{Main\| Diaschismic }}

	The [[period]] of diaschismic is half an [[~~2/1\|~~octave]], and the [[generator]] is a fifth; the [[ploidacot]] is diploid monocot. Three periods gives 1800 cents, and decreasing this by two fifths gives the major third. [[34edo]] is a good tuning choice, with [[46edo]], [[56edo]], [[58edo]], or [[80edo]] being other possibilities. Both [[12edo]] and [[22edo]] support it, and retuning them to a [[mos]] of diaschismic gives two scale possibilities.		The [[period]] of diaschismic is half an [[octave]], and the [[generator]] is a fifth; the [[ploidacot]] is diploid monocot. Three periods gives 1800 cents, and decreasing this by two fifths gives the major third. [[34edo]] is a good tuning choice, with [[46edo]], [[56edo]], [[58edo]], or [[80edo]] being other possibilities. Both [[12edo]] and [[22edo]] support it, and retuning them to a [[mos]] of diaschismic gives two scale possibilities.

	This temperament is also known as '''srutal''' in the 5-limit, but that name more strictly speaking refers to the [[#Srutal\|34d & 46 extension]] to the [[7-limit]] that adds [[4375/4374]] to the comma list.		This temperament is also known as '''srutal''' in the 5-limit, but that name more strictly speaking refers to the [[#Srutal\|34d & 46 extension]] to the [[7-limit]] that adds [[4375/4374]] to the comma list.

Overthink: /* Pajara */ subgroup extensions

2026-06-02T18:39:02Z

Pajara: subgroup extensions

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 === 11-limit ===
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 ==== 13-limit ====
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 ==== 13-limit ====

FloraC: Move srutal archagall to subgroup extensions section as it skips multiple primes. - redundant category

2026-05-29T09:11:18Z

Move srutal archagall to subgroup extensions section as it skips multiple primes. - redundant category

← Older revision		Revision as of 09:11, 29 May 2026
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	{{Mapping\|legend=1\| 2 0 11 \| 0 1 -2 }}		{{Mapping\|legend=1\| 2 0 11 \| 0 1 -2 }}

	: mapping generators: ~45/32, ~3		: mapping generators: ~45/32, ~3

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	==== Subgroup extensions ====		==== Subgroup extensions ====
	Since the diaschisma factors into ([[256/255]])<sup>2</sup>([[289/288]]) in the 17-limit, it extends naturally to the 2.3.5.17 subgroup as ''srutal archagall'', ~~documented right below~~. The [[S-expression]]-based comma list of this temperament is {[[256/255\|S16]], [[289/288\|S17]]}.		Since the diaschisma factors into ([[256/255]])<sup>2</sup>([[289/288]]) in the 17-limit, it extends naturally to the 2.3.5.17 subgroup as ''srutal archagall'', considered in [[#Subgroup extensions]]. The [[S-expression]]-based comma list of this temperament is {[[256/255\|S16]], [[289/288\|S17]]}.

	~~=== Srutal archagall ===~~
	~~{{See also \| Fiventeen }}~~

	~~Subgroup: 2.3.5.17~~

	~~Comma list: 136/135, 256/255~~

	~~Subgroup-val mapping: {{mapping\| 2 0 11 5 \| 0 1 -2 1 }}~~

	~~: mapping generators: ~17/12, ~3~~

	~~Optimal tunings:~~
	* WE: ~45/32 = 599.5585{{c}}, ~3/2 = 704.6188{{c}}
	* CWE: ~45/32 = 600.0000{{c}}, ~3/2 = 705.1356{{c}}

	~~{{Optimal ET sequence\|legend=0\| 10, 12, 22, 34, 80, 114, 194bc }}~~

	~~Badness (Sintel): 0.212~~

	== Septimal diaschismic ==		== Septimal diaschismic ==
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	Mapping: {{mapping\| 2 0 11 12 -1 \| 0 2 -4 -4 5 }}		Mapping: {{mapping\| 2 0 11 12 -1 \| 0 2 -4 -4 5 }}

	: mapping generators: ~2, ~55/32		: mapping generators: ~2, ~55/32

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	{{Mapping\|legend=1\| 4 0 22 43 \| 0 1 -2 -5 }}		{{Mapping\|legend=1\| 4 0 22 43 \| 0 1 -2 -5 }}

	: mapping generators: ~25/21, ~3		: mapping generators: ~25/21, ~3

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	{{Mapping\|legend=1\| 2 1 9 -2 \| 0 2 -4 7 }}		{{Mapping\|legend=1\| 2 1 9 -2 \| 0 2 -4 7 }}

	: mapping generators: ~45/32, ~35/24		: mapping generators: ~45/32, ~35/24

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	{{Mapping\|legend=1\| 2 1 9 11 \| 0 2 -4 -5 }}		{{Mapping\|legend=1\| 2 1 9 11 \| 0 2 -4 -5 }}

	: mapping generators: ~45/32, ~10/7		: mapping generators: ~45/32, ~10/7

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	{{Mapping\|legend=1\| 2 0 11 -15 \| 0 2 -4 13 }}		{{Mapping\|legend=1\| 2 0 11 -15 \| 0 2 -4 13 }}

	: mapping generators: ~45/32, ~140/81		: mapping generators: ~45/32, ~140/81

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	{{Mapping\|legend=1\| 2 0 11 4 \| 0 2 -4 1 }}		{{Mapping\|legend=1\| 2 0 11 4 \| 0 2 -4 1 }}

	: mapping generators: ~45/32, ~7/4		: mapping generators: ~45/32, ~7/4

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	{{Mapping\|legend=1\| 2 1 9 2 \| 0 3 -6 5 }}		{{Mapping\|legend=1\| 2 1 9 2 \| 0 3 -6 5 }}

	: mapping generators: ~45/32, ~9/7		: mapping generators: ~45/32, ~9/7

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	{{Mapping\|legend=1\| 2 2 7 6 \| 0 3 -6 -1 }}		{{Mapping\|legend=1\| 2 2 7 6 \| 0 3 -6 -1 }}

	: mapping generators: ~45/32, ~8/7		: mapping generators: ~45/32, ~8/7

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	{{Mapping\|legend=1\| 2 0 11 8 \| 0 4 -8 -3 }}		{{Mapping\|legend=1\| 2 0 11 8 \| 0 4 -8 -3 }}

	: mapping generators: ~45/32, ~21/16		: mapping generators: ~45/32, ~21/16

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	Badness (Sintel): 1.13		Badness (Sintel): 1.13

			== Subgroup extensions ==
			=== Srutal archagall (2.3.5.17) ===
			{{See also \| Fiventeen }}

			Subgroup: 2.3.5.17

			Comma list: 136/135, 256/255

			Subgroup-val mapping: {{mapping\| 2 0 11 5 \| 0 1 -2 1 }}
			: mapping generators: ~17/12, ~3

			Optimal tunings:
			* WE: ~45/32 = 599.5585{{c}}, ~3/2 = 704.6188{{c}}
			* CWE: ~45/32 = 600.0000{{c}}, ~3/2 = 705.1356{{c}}

			{{Optimal ET sequence\|legend=0\| 10, 12, 22, 34, 80, 114, 194bc }}

			Badness (Sintel): 0.212

	[[Category:Temperament families]]		[[Category:Temperament families]]
	[[Category:Diaschismic family\| ]] <!-- main article -->		[[Category:Diaschismic family\| ]] <!-- main article -->
	~~[[Category:Diaschismic\| ]] <!-- key article -->~~
	[[Category:Rank 2]]		[[Category:Rank 2]]

Lériendil: /* Shrutar */

2026-02-08T16:48:55Z

Shrutar

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	By adding 121/120 or 176/175 to the commas, shrutar can be extended to the 11-limit, which loses a bit of accuracy, but picks up low-complexity 11-limit harmony, making shrutar quite an interesting 11-limit system. 68, 114 or a 14<sup>(1/7)</sup> generator can again be used as tunings.		By adding 121/120 or 176/175 to the commas, shrutar can be extended to the 11-limit, which loses a bit of accuracy, but picks up low-complexity 11-limit harmony, making shrutar quite an interesting 11-limit system. 68, 114 or a 14<sup>(1/7)</sup> generator can again be used as tunings.

			Additionally, shrutar can employ the standard diaschismic mapping of prime 17, and most naturally represents the 2.3.5.7.11.17 subgroup temperament where 15:16:17:18 and 32:33:34:35:36 are equalized. Shrutar canonically maps primes 13, 19, and 23 as the 46 & 68 temperament; these mappings are significantly more complex and need finer tuning, however.

	[[Subgroup]]: 2.3.5.7		[[Subgroup]]: 2.3.5.7
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	Badness (Sintel): 1.16		Badness (Sintel): 1.16

	=== 17-limit ===		==== 17-limit ====
	Subgroup: 2.3.5.7.11.13.17		Subgroup: 2.3.5.7.11.13.17

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	Badness (Sintel): 0.953		Badness (Sintel): 0.953

	=== 19-limit ===		==== 19-limit ====
	Subgroup: 2.3.5.7.11.13.17.19		Subgroup: 2.3.5.7.11.13.17.19

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	Badness (Sintel): 1.07		Badness (Sintel): 1.07

	=== 23-limit ===		==== 23-limit ====
	Subgroup: 2.3.5.7.11.13.17.19.23		Subgroup: 2.3.5.7.11.13.17.19.23

Overthink: /* 7-limit extensions */ note pajara tempers 50/49

2026-01-11T20:25:59Z

7-limit extensions: note pajara tempers 50/49

← Older revision		Revision as of 20:25, 11 January 2026
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	To get the 7-limit extensions, we add another comma:		To get the 7-limit extensions, we add another comma:
	* Septimal diaschismic adds [[126/125]], the starling comma, to obtain 7-limit harmony by more complex methods than pajara, but with greater accuracy.		* Septimal diaschismic adds [[126/125]], the starling comma, to obtain 7-limit harmony by more complex methods than pajara, but with greater accuracy.
	* Pajara ~~derives from~~ [[64/63]] and is a popular and well-known choice.		* Pajara adds [[50/49]] or [[64/63]] and is a popular and well-known choice.
	* Srutal adds [[4375/4374]], the ragisma, which is about as accurate as septimal diaschismic but has a much more complex mapping of 7.		* Srutal adds [[4375/4374]], the ragisma, which is about as accurate as septimal diaschismic but has a much more complex mapping of 7.
	* Keen adds [[875/864]].		* Keen adds [[875/864]].
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	Those all keep the same half-octave period and fifth generator.		Those all keep the same half-octave period and fifth generator.

	Bidia adds [[3136/3125]], the hemimean comma, with a 1/4-octave period. Shrutar adds [[245/243]] and shru adds [[392/375]], with a quartertone generator. Sruti adds [[19683/19600]] and anguirus adds [[49/48]], with a neutral third or hemitwelfth generator. Those split the original generator in two. Echidna adds [[1728/1715]], the orwellisma, with a ~9/7 generator. Echidnic adds [[686/675]], the senga, with a ~8/7 generator. Those split the original generator in three. Finally, quadrasruta adds [[2401/2400]] and splits the original generator in four.		Bidia adds [[3136/3125]], the hemimean comma, with a 1/4-octave period. Shrutar adds [[245/243]] and shru adds [[392/375]], with a quartertone generator. Sruti adds [[19683/19600]] and anguirus adds [[49/48]], with a neutral third or hemitwelfth generator. Those split the original generator in two. Echidna adds [[1728/1715]], the orwellisma, with a ~9/7 generator. Echidnic adds [[686/675]], the senga, with a ~8/7 generator. Those split the original generator in three. Finally, quadrasruta adds [[2401/2400]] and splits the original generator in four.

	==== Subgroup extensions ====		==== Subgroup extensions ====

Overthink: /* Septimal diaschismic */ + a few notes

2026-01-11T20:22:57Z

Septimal diaschismic: + a few notes

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	{{See also\| Srutal vs diaschismic }}		{{See also\| Srutal vs diaschismic }}

	A simpler characterization than the one given by the normal comma list is that septimal diaschismic adds [[126/125]] or [[5120/5103]] to the set of commas, and it can also be called {{nowrap\| 46 & 58 }}. However described, septimal diaschismic has a 1/2-octave period and a sharp fifth generator like the 5-limit version, but not so sharp, giving a more accurate but more complex temperament. [[104edo]] provides an excellent tuning, which is close to tuning [[7/4]] just by making the fifth 703.897 cents.		A simpler characterization than the one given by the normal comma list is that septimal diaschismic adds [[126/125]] or [[5120/5103]] to the set of commas, and it can also be called {{nowrap\| 46 & 58 }}. However described, septimal diaschismic has a 1/2-octave period and a sharp fifth generator like the 5-limit version, but not so sharp, giving a more accurate but more complex temperament. [[104edo]] with the 104c [[val]] provides an excellent tuning, which is close to tuning [[7/4]] just by making the fifth 703.897 cents.

	Diaschismic extends naturally to the 17-limit, for which the same tunings may be used, making it one of the most important of the higher-limit rank-2 temperaments. Adding the 11-limit adds the commas 176/175, 896/891 and 441/440. The 13-limit yields 196/195, 351/350, and 364/363; the 17-limit adds 136/135, 221/220, and 442/441. If you want to explore higher-limit harmonies, diaschismic is certainly one excellent way to do it; [[mos]] of 34 notes and even more the 46-note mos will encompass very great deal of it. Of course 46 or 58 equal provide alternatives which in many ways are similar, particularly in the case of 58.		Diaschismic extends naturally to the 17-limit, for which the same tunings may be used, making it one of the most important of the higher-limit rank-2 temperaments. Adding the 11-limit adds the commas 176/175, 896/891 and 441/440. The 13-limit yields 196/195, 351/350, and 364/363; the 17-limit adds 136/135, 221/220, and 442/441. This mapping can also be rationalized by [[parapyth]], which makes sense due to the sharp fifth, and prime 17 is found as in srutal archagall. If you want to explore higher-limit harmonies, diaschismic is certainly one excellent way to do it; [[mos]] scales of 34 notes and even more the 46-note mos will encompass very great deal of it. Of course 46 or 58 equal provide alternatives which in many ways are similar, particularly in the case of 58.

	[[Subgroup]]: 2.3.5.7		[[Subgroup]]: 2.3.5.7